Circles (A38a), (A38b), (A38c), and (A38d): The CD-circle Powerian Pairs

Definition

Consider the circle (M) on diameter CD, and on this circle the endpoints of the diameter perpendicular to OM. The smallest circles tangent to (O) and to the segments connecting O to the endpoint are Archimedean. These form thus two Powerian pairs. [van Lamoen, 2007]

Generalization

Hiroshi Okumura generalized this cirle considerably in 2013: Consider the circle with center O through C. Let D be on this circle and E a point on the circle (AB) (on diameter AB). Then let (F) be the semicircle on diameter DE and JG the diameter of this circle perpendicular to OF. The smalles circle tangent to (AB) and to the segments OJ and OG are Archimedean. The original Power circles fit in this generalization as well and are found with D=C and, E=A and E=B respectively. [Okumura, 2013].

Source:

Hiroshi Okumura, Lamoenian Circles of the Collinear Arbelos, KoG, vol. 17 nr. 17. page 9--13


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